Quoted from Iona & Peter Opie, The Singing Game (Oxford UP, 1985, p. 218):

The significance of 'weasel', if any, has been a favourite subject for speculation. It has been suggested that a 'weasel' is a tailor's flat-iron, a hatter's tool, a piece of silver plate, or a 'weasel and stoat' (rhyming slang for coat) which must be 'popped' or pawned because of visits to the Eagle; that it is a mishearing of weevil or vaisselle (which leads to further obscurities); that it was James I's nickname ('because of his thin sharp features and red hair) and that, 'rice' and 'treacle' being slang terms for potassium nitrate and charcoal respectively, the rhyme refers to the Gunpowder Plot; that the phrase describes a sixpence expended, a cork being drawn out of a bottle, or the sinuous weasel-like movement of the dancer passing under the arms of his partners. But even when the dance was at the height of its popularity nobody seems to have known what the phrase meant, and W.R. Mandale, in his comic song 'Pop Goes the Weasel', says that after enquiring of everyone he met,

I'm still as wise as e'er I was,
As full's an empty pea-shell,
In as far as the true history goes
Of 'Pop goes the weasel'.

~Masato